The Expurgated Error Exponent is Not Universally Achievable

TL;DR

This paper demonstrates that the expurgated error exponent cannot be universally achieved for all discrete memoryless channels, contrasting with the known universal attainability of the random-coding exponent, and shows limitations of MMI decoding.

Contribution

It proves the non-universality of the expurgated error exponent for DMCs and constructs specific channel families where it cannot be attained.

Findings

Expurgated exponent is not universally attainable for certain DMC families.

MMI decoding fails to achieve the expurgated exponent for these channels.

The random-coding exponent remains universally attainable via MMI decoding.

Abstract

We study the universal attainability of the expurgated error exponent for discrete memoryless channels (DMCs). While the random-coding exponent is known to be universally attainable via maximum mutual information (MMI) decoding for DMCs, it remains open whether the expurgated exponent can be attained universally. We show that this is not the case in general. Specifically, we construct a family of DMCs for which no single sequence of codes can attain the expurgated exponent simultaneously for all channels in the family, even at rate zero. In addition, for the same channel family, we show that MMI decoding fails to achieve the expurgated exponent for any channel in the family.